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Partial weight one modularity for Galois representations associated to mod $p$ Hilbert modular forms

Hanneke Wiersema

Abstract

Let $p$ be an odd prime. Let $ρ: G_F \to \mathrm{GL}_2(\overline{\mathbb{F}}_p)$ be a Galois representation of a totally real field $F$. For a small partial weight one weight $(k,0)$, we prove that modularity of $ρ$ can be characterised using $p$-adic Hodge theory, as conjectured by Diamond and Sasaki. We show that if $ρ$ is modular with respect to a partial weight one mod $p$ Hilbert modular form, then each of its local representations has a crystalline lift with prescribed Hodge--Tate weights. Conversely, if for each $v|p$ the restriction $ρ|_{G_{F_v}}$ has a crystalline lift with certain irregular weights, we show that $ρ$ arises from a partial weight one Hilbert modular form. Our method consists of translating results from regular to irregular weights. We do this globally, relating modularity of regular weights to modularity of irregular weights and vice versa, and also use the local, $p$-adic Hodge theory analogue of this, which is recent work of the author.

Partial weight one modularity for Galois representations associated to mod $p$ Hilbert modular forms

Abstract

Let be an odd prime. Let be a Galois representation of a totally real field . For a small partial weight one weight , we prove that modularity of can be characterised using -adic Hodge theory, as conjectured by Diamond and Sasaki. We show that if is modular with respect to a partial weight one mod Hilbert modular form, then each of its local representations has a crystalline lift with prescribed Hodge--Tate weights. Conversely, if for each the restriction has a crystalline lift with certain irregular weights, we show that arises from a partial weight one Hilbert modular form. Our method consists of translating results from regular to irregular weights. We do this globally, relating modularity of regular weights to modularity of irregular weights and vice versa, and also use the local, -adic Hodge theory analogue of this, which is recent work of the author.
Paper Structure (10 sections, 11 theorems, 38 equations)

This paper contains 10 sections, 11 theorems, 38 equations.

Table of Contents

  1. Introduction
  2. Mod $p$ Hilbert modular forms, weight shifting operators and geometric modularity
  3. Mod $p$ Hilbert modular forms and their Galois representations
  4. Properties of Hilbert modular eigenforms
  5. Partial Hasse invariants and partial $\Theta$ operators
  6. Geometric modularity
  7. Partial weight one modularity
  8. The regular weights
  9. Statement and proof of the theorem
  10. Main result

Key Result

Theorem 1.1

Let $F$ be a totally real field in which $p$ is unramified, and suppose $p$ is odd. Suppose that $\rho:G_F \to \mathop{\rm GL}\nolimits_2({\overline{\mathbb F}_p})$ is irreducible and modular. Suppose that for $\rho$ the Buzzard--Diamond--Jarvis conjecture holds and that for regular weights algebrai

Theorems & Definitions (28)

  • Theorem 1.1: Theorem \ref{['locglobviceversa']}
  • Definition 1.2
  • Definition 2.1
  • Lemma 2.2: DS
  • Theorem 2.3: DiamondKassaei.
  • Theorem 2.4
  • Remark 2.5
  • Theorem 2.6
  • proof
  • Definition 3.1
  • ...and 18 more